Vacuum Polarization and Photon Propagation in an Electromagnetic Plane Wave

The QED vacuum polarization in external monochromatic plane-wave electromagnetic fields is calculated with spatial and temporal variations of the external fields being taken into account. We develop a perturbation theory to calculate the induced electromagnetic current that appears in the Maxwell equations, based on Schwinger's proper-time method, and combine it with the so-called gradient expansion to handle the variation of external fields perturbatively. The crossed field, i.e., the long wavelength limit of the electromagnetic wave is first considered. The eigenmodes and the refractive indices as the eigenvalues associated with the eigenmodes are computed numerically for the probe photon propagating in some particular directions. In so doing, no limitation is imposed on the field strength and the photon energy unlike previous studies. It is shown that the real part of the refractive index becomes less than unity for strong fields, the phenomenon that has been known to occur for high-energy probe photons. We then evaluate numerically the lowest-order corrections to the crossed-field resulting from the field variations in space and time. It is demonstrated that the corrections occur mainly in the imaginary part of the refractive index.

The QED vacuum polarization in external monochromatic plane-wave electromagnetic fields is calculated with spatial and temporal variations of the external fields being taken into account. We develop a perturbation theory to calculate the induced electromagnetic current that appears in the Maxwell equations, based on Schwinger's proper-time method, and combine it with the so-called gradient expansion to handle the variation of external fields perturbatively. The crossed field, i.e., the long wavelength limit of the electromagnetic wave is first considered. The eigenmodes and the refractive indices as the eigenvalues associated with the eigenmodes are computed numerically for the probe photon propagating in some particular directions. In so doing, no limitation is imposed on the field strength and the photon energy unlike previous studies. It is shown that the real part of the refractive index becomes less than unity for strong fields, the phenomenon that has been known to occur for high-energy probe photons. We then evaluate numerically the lowest-order corrections to the crossed-field resulting from the field variations in space and time. It is demonstrated that the corrections occur mainly in the imaginary part of the refractive index.

Introduction
In the quantum vacuum, virtual particles and anti-particles are produced and annihilated repeatedly in very short times as intuitively represented by bubble Feynman diagrams. When an external field is applied, even these virtual particles are affected, leading to modifications of the property of quantum vacuum. One of the interesting consequences is a deviation of the refractive index from unity accompanied by a birefringence, i.e., distinct refractive indices for different polarization modes of photon 1 . It is a purely quantum effect that becomes remarkable when the strength of the external field approaches or even exceeds the critical value, f c = m 2 /e with m and e being the electron mass and the elementary charge, respectively, whereas, the deviation of the refractive index from unity is proportional to the field-strength squared for much weaker fields. Photon splitting, which is another phenomenon in external fields, has been also considered [2]. Such strong electromagnetic fields are not unrealistic these days. In fact, the astronomical objects called magnetars are a subclass of neutron stars, which are believed to have dipole magnetic fields of ∼ 10 14−15 G [3] 2 . Although the origin of such strong magnetic fields is still unknown, they are supposed to have implications for various activities of magnetars such as giant flares and X-ray emissions [4]. In fact, their strong magnetic fields are thought to affect the polarization properties of surface emissions from neutron stars by the quantum effect [5,6]. This phenomenon may have indeed been detected in a recent optical polarimetric observation [7]. The quantum correction may also play an important role through the socalled resonant mode conversions [8][9][10]. On the other hand, the progress in the high-field laser is very fast. Although the highest intensity realized so far by Hercules laser at CUOS [11] is still sub-critical (2 × 10 22 W/cm 2 ) for the moment, we may justifiably expect that the laser intensity will reach the critical value in not-so-far a future. Some theoretical studies on the vacuum polarization are meant for the experimental setups in the high-field laser [12][13][14][15][16].
The study of the vacuum polarization in strong-field QED has a long history. It was pioneered by Toll in 1952 [17]. He studied in his dissertation the polarization of vacuum in stationary and homogeneous magnetic fields in detail and many authors followed with different methods, both analytic and numerical [18][19][20][21][22][23][24][25][26][27][28], and obtained the refractive indices. The vacuum polarization for mixtures of constant electric and magnetic fields was also investigated [29][30][31][32][33][34]. Note that such fields can be brought to either a purely magnetic or a purely electric field by an appropriate Lorentz transformation, with so-called crossed fields being an exception.
In [17], the polarization in the crossed field was also discussed. The crossed field may be regarded as a long wavelength limit of electromagnetic waves, having mutually orthogonal electric and magnetic fields of the same amplitude. Toll first calculated the imaginary part of the refractive index from the amplitude of pair creations and then evaluated the real part of refractive index via the Kramers-Kronig relation. Although there was no limitation to the probe-photon energy, the external-field strength was restricted to small values (weak-field limit) because he ignored the modification of the dispersion relation of the probe photon. Baier and Breitenlohner [35] obtained the refractive index for the crossed field in two different ways: they first employed the polarization tensor that had been inferred in [18] from the 1loop calculation for the external magnetic fields, and utilized in the second method the expansion of the Euler-Heisenberg Lagrangian to the lowest order of field strength. Note that both approaches are valid only for weak fields or low-energy probe photons.
The expression of the polarization tensor to the full order of field strength for the external crossed field was obtained from the 1-loop calculation with the electron propagator derived either with Schwinger's proper-time method [36] or with Volkov's solution [37]. In [36], the general expressions for the dispersion relations and the refractive indices of the two eigenmodes were obtained. Note, however, that the refractive indices were evaluated only in the limit of the weak-field and strong-field 3 . On the other hand, another expression of polarization tensor was obtained and its asymptotic limit was derived in [37] although the refractive index was not considered.
The evaluation of the refractive index based on the polarization tensor of [37] was attempted by Heinzl and Schröder [38] in two different ways: the first one is based on the hypothesized expression of the polarization tensor in the so-called large-order expansion with respect to the probe-photon energy; in the evaluation of the real part of the refractive index, the external crossed field was taken into account only to the lowest order of the field strength in each term of the expansion and the imaginary part was estimated from the hypothesized integral representation; in the second approach, the polarization tensor was expanded with respect to the product of the external-field strength and the probe-photon energy, and the refractive index was evaluated; the imaginary part was calculated consistently to the leading order and the anomalous dispersion for high-energy probe photons, which had been demonstrated by Toll [17], was confirmed. Note that in these evaluations of the refractive index in the crossed field, the modification of the dispersion relation for the probe photon was again ignored as in [17] and hence the results cannot be applied to super-critical fields.
It should be now clear that the vacuum polarization and the refractive index have not been fully evaluated for supra-critical field strengths even in the crossed field. One of our goals is hence to do just that.
It is understandable, on the other hand, that the evaluation of the refractive index in the external electromagnetic plane-wave is more involved because of its non-uniformity. In fact, the refractive index has not been obtained except for some limiting cases. The polarization tensor and the refractive index in the external plane-wave were first discussed by Becker and Mitter [39]. They derived the polarization tensor in momentum space from the 1-loop calculation with the electron propagator obtained by Mitter [40], which is actually Volkov's propagator represented in momentum space. Although the formulation is complete, the integrations were performed only for circularly polarized plane-waves as the background. The refractive indices were then evaluated at very high energies ( m) of the probe photon.
Baier et al. [41] calculated scattering amplitudes of a probe photon again by the circularlypolarized external plane-wave to the 1-loop order, employing the electron propagator expressed with the proper-time integral. The general expression of the dispersion relation was obtained but evaluated only in the weak-field and low-energy limit. The refractive indices for the eigenmodes of probe photons were also calculated in this limit alone. Affleck [42] treated this problem by expanding the Euler-Heisenberg Lagrangian to the lowest order of the field strength, assuming that the external field varies slowly in time and space. The refractive index was evaluated only in the weak-field limit again. Recently, yet another representation of the polarization tensor in the external plane-wave was obtained from the calculation of the 1-loop diagram with Volkov's electron propagator [43]. Only the expression of the polarization tensor was obtained, however, and no attempt was made to evaluate it in this study.
In their paper [13], Dinu et al. employed the light front field theory, one of the most mathematically sophisticated formulations, to derive the amplitude of photon-photon scatterings, from which the refractive index integrated over the photon path was obtained. They calculated it for a wide range of the probe-photon energy and field strength. Although they gave the expression for the local refractive index, it was not evaluated. The eigenmodes of probe photons were not calculated, either. 3/50 In this paper, we also derive the expression of the polarization tensor and the refractive index for the external electromagnetic plane-wave, developing a perturbation theory for the induced electromagnetic current based on the proper-time method. It is similar to Adler's formulation [20] but is more general, based on the interaction picture, or Furry's picture, and not restricted to a particular field configuration. Combining it with the so-called gradient expansion, we calculate the lowest-order correction from temporal and spatial field variations to the induced electromagnetic current, and hence to the vacuum polarization tensor also, for the crossed fields. This is nothing but the WKB approximation and, as such, may be applicable not only to the electromagnetic wave but also to any slowly-varying background electromagnetic fields. We then evaluate numerically the refractive indices for eigenmodes of the Maxwell equations with the modification of the dispersion relation being fully taken into account. Note that unlike [13] our results are not integrated over the photon path but local, being obtained at each point in the plane wave.
The paper is organized as follows: we first review Schwinger's proper time method briefly and then outline the perturbation theory based on the Furry picture to obtain the induced electromagnetic current to the linear order of the field strength of the probe photon in Sec. 2. This is not a new stuff. We then apply it to the plane-wave background in Sec. 3; in so doing, we also appeal to the so-called gradient expansion of the background electromagnetic wave around the crossed field. Technical details are given in Appendices. Numerical evaluations are performed both for the crossed fields and for the first-order corrections in Sec. 4; we summarize the results and conclude the paper in Sec. 5.

Perturbation Theory in Proper-Time Method
In this section, we briefly summarize Schwinger's proper-time method and outline its perturbation theory, which will be applied to monochromatic plane-waves in the next section.

Schwinger's Proper-Time Method
The effective action of electromagnetic fields is represented as where Γ cl is the classical action and Γ q is the quantum correction, which satisfies the following relation: Then, the vacuum Maxwell equation is modified as Although there is no electromagnetic current generated by real charged particles in the vacuum, j µ defined in this way is referred to as the induced electromagnetic current [33]. This term can be written with the electron propagator G(x, y) [44] with tr in Eq. (2) being the trace, or the diagonal sum on spinor indices; γ µ 's are the gamma matrices. In this paper, the Greek indices run over 0 through 3 and the Minkowski metric is assumed to be η = diag(+, −, −, −).
The electron propagator G in the external electromagnetic field is different from the ordinary one in the vacuum and the modification by the external field, the strength of which is 4/50 close to or even exceeds the critical value f c , cannot be treated perturbatively. The propertime method is a powerful tool to handle such situations. The electron propagator satisfies the Dirac equation in the external electromagnetic field A µ : It is supposed in the proper-time method that there exists an operatorĜ, the xrepresentation of which gives the propagator as x|Ĝ|y = G(x, y). Then Eq. (4) can be cast into the following equation for the operators: where1 is the unit operator andΠ µ = i∂ µ − eA µ . Here we used δ 4 (x − y) = x|y . From this equation, the operatorĜ is formally solved aŝ which can be cast into the following integral form: In the above expression, the parameter s is called the proper-time and −iε is introduced to make the integration convergent as usual and will be dropped hereafter for brevity. The electron propagator, being an x-representation of this operator, is obtained as 2 )s |y .
Here we had better comment on the boundary condition for the electron propagator, or the causal Green function, in the electromagnetic wave. This issue may be addressed most conveniently for finite wave trains in the so-called light front formulation (e.g. [45,46]), in which double null coordinates are employed. This is because the asymptotic states in the remote past and future (in the null-coordinate sense) are unambiguously defined [46], which is crucially important particularly when one calculates S-matrix elements [13,46]; it is also important that the translational symmetry is manifest in one of the null coordinates. Then the causal Green function is obtained in the usual way, i.e., by the appropriate linear combination of the homogeneous Green functions with positive-and negative-energies according to the time ordering in the null coordinate [45,46]. On the other hand, it is a well-known fact that the Dirac equation can be solved in a closed form for an arbitrary plane wave [40,47]. It is then possible to construct the same causal Green function with these Volkov solutions [37,40]. According to Ritus [37], all that is needed is a well-known −i prescription, i.e., the introduction of an infinitesimal negative imaginary mass. It was pointed out by Mitter [40] then that this is equivalent to the same prescription in the proper-time method of Schwinger, that is, the formulation we adopt in this paper (see Eq. (7)). In this sense, the propagator we employ in this paper is the causal Green function thus obtained in the limit of the infinite wave train. As will become clear later (see Eq. (38) in Section 3), since we employ the gradient expansion in the local approximation, the distinction between the finite or infinite wave train will not be important in our formulation.
Returning to Eq. (8) and interpreting the operator e −i(−(γ µΠ µ ) 2 )s as the evolution operator in the proper-time, one can reduce the original field-theoretic problem to the one in quantum 5/50 mechanics for the Hamiltonian H = −(γ µΠ µ ) 2 . Then the transformation amplitude is given as 2 )s |y = x|e −iHs |y = x(s)|y(0) .
Here the state |x(s) is defined as the eigenstate for the operatorx in the Heisenberg picture: The Hamiltonian H is expressed as where we used the Clifford algebra for the gamma matrices {γ µ , γ ν } = 2η µν and the commutation relation [Π µ , Π ν ] = −ieF µν to obtainΠ 2 =Π µΠ µ and σ µν = i 2 [γ µ , γ ν ]; A µ and F µν are the vector potential and the field tensor for the external electromagnetic field, respectively. The proper-time evolutions of the operatorsx andΠ are given by the Heisenberg equations: Then, the induced electromagnetic current j µ in Eq.
(2) is represented as follows [20]: Note that this is equivalent to the 1-loop approximation with the external field being fully taken into account. In order to obtain the refractive index of the vacuum in the presence of an external electromagnetic field, we have to consider a probe photon in addition to the background electromagnetic field and apply Eq. (3) to the amplitude of the probe photon. In so doing, the induced electromagnetic current j µ needs to be evaluated to the linear order of the amplitude of the probe photon and the perturbation theory is required at this point [33]. The Heisenberg equations given above can be solved analytically for some limited cases such as time-independent homogeneous electric or magnetic fields and single electromagnetic plane-waves [44]. We will employ the latter as an unperturbed solution in the perturbative calculations in Section 3. It is stressed that calculating the effective action for a given plane-wave background and taking its derivative with respective to the field strength is not sufficient for the evaluation of the refractive index, since the probe photon in general has a different wavelength and propagates in a different direction from those of the background electromagnetic wave. We hence need to take these differences fully into account in the perturbative calculations of the induced electromagnetic current. This was essentially done by [39] in a different framework, i.e., performing 1-loop calculations in momentum space. In this paper we assume that the background wave has a long wavelength and calculate the refractive index locally in the sense of the WKB-approximation. In so doing, we appeal to the gradient expansion of the background plane wave as explained in Section 3. 6/50

Outline of Perturbation Theory
We now consider the perturbation theory in the proper time method. The purpose is to evaluate the induced electromagnetic current Eq. (14) up to the linear order of the amplitude of the probe photon, which is supposed to propagate in an external electromagnetic field. It is then plugged into Eq. (3) to derive the refractive indices. The Heisenberg equations (12), (13) can be analytically solved for a single monochromatic electromagnetic plane-wave [44]. The calculation of the first order corrections to this solution is the main achievement in this paper. As explained in the next section, we employ further the gradient expansion of the background electromagnetic wave, which in turn enables us to obtain the refractive indices locally in the WKB sense. In this section, we give the outline of the generic part of this perturbation theory, which is not limited to the plane-wave background. We will then proceed to its application to the monochromatic plane-wave background in the next section.
In the perturbation theory, the external electromagnetic fields are divided into two pieces: the background A µ and the perturbation b µ . The corresponding field strengths are denoted by F µν and g µν , respectively. We take the latter into account only to the first order. Then, the Hamiltonian given in Eq. (11) can be written as In this expression, H (0) is the unperturbed Hamiltonian, for which we assume that the proper-time evolution is known, preferably analytically as in the time-independent homogeneous electric or magnetic fields and the single plane-wave. δH is the perturbation to the Hamiltonian. It is evaluated to the first order of b µ and expressed with δΠ µ = −eb µ as In the proper-time method, the amplitudes of operators such as x(s)|Π µ (s)|x(0) are evaluated very frequently and in the perturbation theory they need to be calculated with perturbations to both the operators and the states being properly taken into account. In so doing, we employ the interaction picture, which is also referred to as the Furry picture [46] in the current case, rather than making full use of the properties of particular field configurations as in [20]. The relation between the operator in the Heisenberg pic-tureÂ H (u) and that in the interaction pictureÂ I (u) is then given by the transformation: It also satisfies the following equation: Here the perturbation Hamiltonian in the interaction picture δH I is given as 7/50 Note that the right hand side of this equation includes only unperturbed quantities, since the operators obey the free Heisenberg equations in the interaction picture. The transformation amplitude x(s)|x(0) is also expressed with the unperturbed operators and states as In this expression, the index (0) attached to the state indicates its proper-time evolution by H (0) : Since we assume that the interaction and Heisenberg pictures are coincident with each other at u = 0, we have The operatorsΠ(u) and the states |x(u) at an arbitrary proper-time u are expressed with the unperturbed counterpartsΠ (0) ,x (0) and |x (0) via the operator U (u) given in Eq. (17) in a similar way. The amplitudes that appear in Eq. (14) for j µ (x) can be represented as with the operators and states in the interaction picture. The calculations of these amplitudes are accomplished by the permutations of operatorsx(s) andx(0) with the employment of their commutation relations so thatx(s) should sit always to the left ofx(0).

Application to the Single Plane-Wave
In this section, we apply the perturbation theory outlined above to the calculation of the induced electromagnetic current in the monochromatic plane-wave. It is stressed that the distinction between the electromagnetic wave train having a finite or infinite length is not important in our calculations, since they employ only local information of the electromagnetic wave in the background thanks to the gradient expansion. We first summarize the well-known results for the unperturbed background [44]. The plane wave is represented as with f µν being a constant tensor that sets the typical amplitude of the wave and F (Ωξ) being an arbitrary function of Ωξ = Ωn µ x µ ; Ω is a frequency of the wave and n µ is a null vector that specifies the direction of wave propagation. The Heisenberg equations are written in this case as Note that the phase ξ(s) = n µx µ (s) in these equations is an operator and a function of the proper time s. The term that contains ∂ ν F µ ν vanishes in the equation ofΠ because it is 8/50 written as ∂ ν F µ ν = f µ ν n ν [dF (Ωξ(s))/dξ(s)] and the following relation f µ ν n ν = 0 holds for the plane-wave.
To solve these equations, one introduces which one can show is a constant of motion. In this expression, f 2 = f µν f ν λ /n µ n λ is the amplitude squared of the plane-wave and A(Ωξ(s)) is defined as a quantity that satisfies the following relation: F (Ωξ(s)) = dA(Ωξ(s))/dξ(s). Then, the operatorsΠ µ (s) andΠ µ (0) are obtained as follows: C µ is also expressed as The amplitude x (s)|x (0) is given, on the other hand, as To derive the induced electromagnetic current j µ , we use the amplitude x(s)|x(0) , which is immediately obtained from the above equation as We then find from Eqs. (27) through (30) that j µ in Eq. (14) is vanishing as pointed out first by Schwinger in his seminal paper [44]. This situation changes, however, if another plane wave is added.
In this paper, we consider the propagation of a probe photon through the external monochromatic plane-wave with the former being treated as a perturbation to the latter as usual. We have in mind its application to high-field lasers. Since the wavelengths of these lasers are close to optical wavelengths, we assume in the following that the wavelength of the unperturbed monochromatic plane-wave is much longer than the electron's Compton wavelength, or Ω 0 /m 1 for the wave frequency Ω 0 . It may be then sufficient to consider temporal and spatial variations of the unperturbed plane-wave to the first order of Ω 0 . This is equivalent to the so-called gradient expansion of the unperturbed field to the first order, which can be expressed generically as F µν f µν (1 + Ωξ). In fact, the plane-wave field given as ) to the first order. This can be recast into F µν f µν (1 + Ωn α x α ) after shifting coordinates by x α 0 and employing the local amplitude and the gradient of the background field at x α as f µν = f 0µν sin(Ω 0 n α x α 0 ) and Ω = cos(Ω 0 n β x β 0 )Ω 0 , respectively. Note that the above assumption on Ω 0 implies Ω/m 1. Gusynin and Shovkovy [48] developed a covariant formulation to derive the gradient expansion of the QED effective Lagrangian, employing the world-line formalism under the Fock-Schwinger gauge. Although their method is systematic and elegant indeed, the results obtained in their paper cannot be applied to the problem of our current interest, since the actual calculations were done only for the following field configurations: is an arbitrary slowly-varying function of x α while f µν is a constant tensor; the former gives a field variation in space and time and the latter specifies a field configuration. Although it appears quite generic, it does not include the configurations of our concern, i.e., those consisting of two electromagnetic waves propagating in different directions, unfortunately. Note that if the background and probe plane-waves are both traveling in the same direction and having the identical polarization, then one may regard the sum of their amplitudes as Φ and apply the gradient expansion of Gusynin and Shovkovy [48] to them; in this case, however, Schwinger [44] already showed that there is no quantum correction to the effective Lagrangian.
In our method, the probe photon, which is also treated as a classical electromagnetic wave, is assumed to be monochromatic locally. Strictly speaking, it has neither a constant amplitude nor a constant frequency because the external field changes temporally and spatially. As long as the wavelength of the external field is much longer than that of the probe photon, which we assume in the following, the above assumption that the probe field can be regarded as monochromatic locally may be justified. We need to elaborate on this issue a bit further, though. As Becker and Mitter developed in their paper [39], the polarization tensor Π µν (x 1 , x 2 ) depends not only on the difference of the two coordinates x 1 − x 2 but also on each of them separately and, as a result, its Fourier transform has two momenta corresponding to these coordinates. If the electromagnetic wave in the background is monochromatic, then the Floquet theorem dictates that the difference between them should be equal to some multiple of the wave vector of the electromagnetic wave in the background [49]. It follows then that eigenmodes of the probe photon are not diagonal in momentum in general. In fact, they should satisfy the following Maxwell equation: Becker and Mitter Fourier-transformed this equation and attempted to solve it in momentum space. Although they showed analytically that the momenta of probe-photon were indeed mixed in the expected way, they ignored the mixing in actual evaluations of the refractive index, since the effect is of higher order in the coupling constant. We take another approach in this paper. Assuming, as mentioned above, that the electromagnetic wave in the background varies slowly in time and space and hence the probe photon can "see" the local field strength and its gradient alone, we expand the above equation in the small gradient. In so doing, we employ the Wigner representations of variables: where X = (x 1 + x 2 )/2 is the center-of-mass coordinates and Π µν and b µ are regarded in these equations as functions of the relative coordinates x 1 − x 2 and X instead of x 1 and x 2 . Inserting these expressions into the right hand side of Eq. (32) and Fourier-transforming it with respective to the relative coordinates x 1 − x 2 , we obtain in which ∂ Π p is a partial derivative with respective to p acting onΠ; other ∂'s should be interpreted in similar ways; the juxtapositions of two ∂'s stand for four-dimensional contractions; the last expression is the approximation to the lowest order with respective to the gradient in X, which is justified by our assumption. In deriving the third line of the above equations, we employ the following relations: Fourier-transforming the left hand side of Eq. (32) also with respective to the relative coordinates x 1 − x 2 , we obtain finally the "local" Maxwell equation as follows: Note that we also ignore the derivative with respect to X in the kinetic part of the Maxwell equation, which is again valid under the current assumption. We then consider the dispersion relation for the probe photon in the point-wise fashion, plugging the polarization tensor obtained locally this way. This is nothing but the WKB approximation for the propagation of probe photon. Note that the momentum of the probe photon is hence not the one in the asymptotic states [13] but the local one defined at each point in the background electromagnetic wave. It should be also stressed that the derived refractive index is a local quantity.
Although such a quantity may not be easy to detect in experiments, this is regardless the main accomplishment in this paper. Fig. 1 Schematic picture of the system considered in this paper. The external field consists of a non-uniform electric and magnetic fields denoted by E and B, respectively, and the probe photon.
We now proceed to the actual calculations. The induced electromagnetic current is written as In this expression, we drop for brevity the superscript (0) , which means the unperturbed states, and the subscript I , which stands for the operators in the interaction picture. We use these notations in the following. The proper-time evolution operator U (s) is given as for the present case. Note that σ αβ (u) is a proper-time-dependent operator in the interaction picture, which is defined as 12/50 The explicit expression of σ αβ (u) can be easily obtained to the first order of perturbation for the current Hamiltonian evaluated at u = 0 4 as H = −Π 2 (0) + 1 2 eσ µν F µν (0): where we employ the abbreviation (σf ) ≡ σ µν f µν . Note that although ξ(0) does not have a spinor structure and commutes with (σg) ≡ σ αβ g αβ , ξ(s) may have a nontrivial spinor structure induced by the proper-time evolution.
The amplitudes of x(s)|Π µ (s)U (s)|x(0) and x(s)|U (s)Π µ (0)|x(0) can now be expressed with the unperturbed operators and states. The calculations are involved, though, and given in Appendix A. The final expressions are given as 4 The Hamiltonian is proper-time-independent and can be evaluated at any time. 13/50 Then the induced electromagnetic current can be calculated by inserting these amplitudes in Eq. . After these permutations, various amplitudes can be easily obtained from the following relations: x(s)|ξ(s) = ξ x(s)|, In deriving Eqs. (42)-(44), we consider only the neighborhood of the coordinate origin, the linear size of which is much shorter than the wavelength of the background plane-wave but larger than the wavelength of the probe photon. As mentioned earlier, however, the origin is arbitrary and one can shift the coordinates so that the point of interest should coincide with the origin. Hence the results are actually applicable to any point. More discussions on this point will be found in Appendix C. Note also that Furry's theorem dictates that the number of external fields that appear in the expression of the induced electromagnetic current should be even, the details of which can be found in Appendix D.
After all these considerations and calculations, the induced electromagnetic current is given to the lowest order of the perturbation and Ω. The details are presented in Appendix E. Since 14/50 the induced electromagnetic current j µ is vanishing in the absence of the probe photon, it is generated by its presence and is should be proportional to it: Note that we employ the local approximation here again. In this expression, the probe photon is given as b ν exp(−ik α x α ) and the proportionality coefficient Π µ ν is nothing but the polarization tensor at each point. Following Ritus [37], we decompose the polarization tensor so obtained as with three mutually orthogonal vectors wheref µν = ε µνρσ f ρσ /2 is the dual tensor of f µν . Here ε µνρσ is the Levi-Civita antisymmetric symbol, which satisfies ε 0123 = 1. The following abbreviations (kk) = k µ k µ and (kf f k) = k µ f µ ν f ν λ k λ are also used [37]. Then the coefficients are given as follows: where (Ωkn) = Ωk µ n µ is the inner product of the momentum vectors of the external planewave and the probe photon. The refractive indices for physical modes are related to Π 1 and Π 2 . The proper-time integration in Eq. (50) or its pre-decomposition form, Eq. (E1), has to be done numerically. The original form is not convenient for this purpose and we rotate the integral path by −π/3 in the complex plane so that the integral could converge exponentially as s goes to infinity. Note that the rotation angle is arbitrary as long as it is in the range of (0, −π/3]. The refractive index is then obtained by solving the Maxwell equation reduced in the following form: with A µ ν = −(kk)δ µ ν + k µ k ν + Π µ ν . The probe photon is hence described as a non-trivial solution of this homogeneous equation and its dispersion relation is obtained from the relation detA = 0. Note that not all of them are physical. Unphysical modes are easily eliminated, however, by calculating the electric and magnetic field strengths, which are gauge-invariant. It is then found that only two of them associated with Π 1 and Π 2 are physical as expected.
Note that the four momenta of the probe photon thus obtained are no longer null in accordance with the refractive indices different from unity. The polarization vectors are also obtained simultaneously.
In the next section, we show the results of some numerical evaluations. As representative cases, we consider four propagating directions of the probe photon as summarized in Table 1. Since the background plane-wave is assumed to have a definite propagation direction (xdirection) and linear polarization (y-direction), these four directions are not equivalent. For each propagation direction, there are two physical eigenmodes, as mentioned above, which 16/50 Table 1 Eigenmodes of the probe photons with different 4-momenta probe momentum k µ eigenmode mode name (k 0 , k 1 , 0, 0) (0, 0, 1, 0) µ x2 mode (0, 0, 0, 1) µ x3 mode

Results
In this section, we numerically evaluate the refractive index N , which is defined as N = |k|/k 0 . Firstly, the crossed fields are considered and then the first order correction δN in the gradient expansion is calculated for the plane-wave field. The eigenmodes of the probe photon depend on the propagation direction as already mentioned. The refractive index is complex in general with the real part representing the phase velocity of the probe photon divided by the light speed and the imaginary part indicating the decay, possibly via electronpositron pair creations. Since the deviation of the refractive index from unity is usually much smaller than unity, only the deviations are shown in the following: Re[N − 1] and Im [N ]. Note that for all cases considered in this paper, the refractive indices, both real and imaginary parts, of the y1 and z2 modes are identical and so are those of the y3 and z1 modes. Although the exact reason for this phenomenon is not known to us for the moment, the following should be mentioned: the polarization tensor Π µν is expressed as the sum of three contributions proportional to (f k) µ (f k) ν , (f k) µ (f k) ν and G µ G ν given as Eq. (50); each pair of the modes that have the identical refractive index are actually eigenmodes of either (f k) µ (f k) ν or (f k) µ (f k) ν . We will show these degenerate modes with the same color in figures hereafter.

Crossed Fields
As mentioned in Introduction, the vacuum polarization in the crossed fields was already obtained by many authors. The refractive index was also evaluated both analytically and numerically [17,[35][36][37][38]. The regions in the plane of the field strength f and the probephoton energy k 0 that have been investigated in these papers are summarized in Fig. 2. It is apparent from the figure that there is still an unexplored region, which is unshaded. And that is the target of this paper. The parameter ranges we adopted in this paper are displayed in orange in the same figure: we first calculate the refractive index for the external field of the critical value to validate our formulation by comparing our results with those in the previous studies; then we vary the strength of the external field. 17/50 Fig. 2 Regions in the plane of the strength of external crossed fields and the probephoton energy that have been already explored. Each region is labeled as follows: region (1) is the weak-field limit f /f c 1 [17]; region (2) is for the weak-field or low-energy limit e 2 k µ f µ ν f νλ k λ /m 6 1 [35][36][37]; region (3) corresponds to the strong-field and high-energy limit 1 e 2 k µ f µ ν f νλ k λ /m 6 (k 0 /m) 6 × α −3 studied in [36], where α = e 2 /4π is the finestructure constant; region (4) is the region that satisfies (f /f c ) 1 and (f /f c ) × (k 0 /m) 1 explored in [38]. The orange lines indicate the regions, in which the refractive indices are computed numerically in this paper. Note that our method can treat the whole region in this figure in principle.
The polarization tensor Π µ ν in the crossed field is obtained by simply taking the limit of Ω → 0 in Eq. (50). Setting the strength of the external field to the critical value f /f c = 1, we compute the refractive indices for the range of 0.01 ≤ k 0 /m ≤ 1000 6 . Note that the lowenergy regime (k 0 /m 1) has been investigated already as shown in Fig. 2. The real part Re[N − 1] is shown in Fig. 3 with colors indicating different modes of the probe photon. It is found that the deviation of the refractive index from unity is of the order of 10 −4 . As k 0 /m gets smaller, the refractive index approaches the values in the weak-field or low-energy limit (region (2) in Fig. 2), which are written as for the x2 and x3 modes, respectively, where κ 2 = e 2 k µ f µ ν f νλ k λ /m 6 = e 2 f 2 (k 0 + k 1 ) 2 /m 6 is the product of the probe photon energy and the field strength normalized by the critical 6 Shore studied the refractive index of super-critical magnetic fields for a wider range of the photon energy [24]. The results are similar to ours for the crossed field. Here N is a refractive index. We set f /f c = 1. Colors specify different modes. value. Then the typical value of N x2 − 1 can be estimated as

18/50
where I = f 2 /4π is the intensity of the plane wave. The results are hence in agreement with what was already published in [35][36][37][38]. The refractive indices depend on the propagation direction of the probe photon: the modulus |Re[N − 1]| is larger for the photon propagating in the opposite direction to the background plane-wave (the x mode) than those going perpendicularly (the y/z modes); the s mode that propagates obliquely lies normally in between although the modulus is greater for the s3 mode than for the x2 mode. The photons polarized in the z-direction have larger moduli in general except the z mode, which propagates 19/50 in this direction, has a greater modulus when it is polarized in the x-direction. These trends are also true for other results obtained below in this paper.
As k 0 /m becomes larger than ∼ 10, all the refractive indices for different propagation directions appear to converge to unity, which is consistent with [17,38]. This is more apparent in Fig. 4, which zooms into the region of 3 k 0 /m ≤ 1000. It is also seen in the same figure that Re[N − 1] is negative and the modulus |Re[N − 1]| decreases for k 0 /m 10. This trend is consistent with the high-energy limits given in [36] (region (3) in Fig. 2), which are written as for the x2 and x3 modes, respectively. In our formulation, these results are reproduced by putting e −im 2 s to unity and setting (kk) = k 2 0 − k 2 1 equal to zero in Eqs. (52), (53) and (54) for the polarization tensor Π µ ν or Eqs. (E57) and (E59) for the induced electromagnetic current j µ . Note, however, that our numerical results for Re [N − 1] are not yet settled to the asymptotic limits with deviations of ∼ 10% still remaining at k 0 /m ∼ 1000. In this figure, the high energy limits for the x2 and x3 modes are displayed as the lines labeled as ax2 and ax3, respectively. The imaginary parts, on the other hand, have already reached the asymptotic limits at k 0 /m ∼ 1000 (see below). Toll [17] pointed out that unless the Poynting vectors of the probe photon and the external field are parallel to each other, an appropriate Lorentz transformation makes them antiparallel and, as a result, the refractive index depends only on the reduced field strength f rd 20/50 as long as the field strength is not much larger than the critical value. Here θ is the angle between the Poynting vectors of the probe photon and the external field. We hence redraw Fig. 3 as Fig. 5 in the range of 0.1 ≤ k 0 /m ≤ 10 after adjusting the external-field strength so that f rd = f c for all the modes. As expected, the x3, s3, y3 and z1 modes become identical, which is also true for the x2, s2, y1 and z2 modes. The relation also holds for the imaginary part. It is important that these relations are obtained as a result of separate calculations for different propagation directions in our formulation, the fact that guarantees the correctness of our calculations.   The imaginary part of the refractive index Im[N ] is shown in Fig. 6 for the same case. It is found that the imaginary part is non-vanishing down to k 0 = 0 although it diminishes 21/50 very rapidly for k 0 /m 0.1. It is also seen that Im[N ] for each photon mode reaches its maximum at k 0 /m ∼ 1 and it decreases monotonically for higher energies. These behaviors are also consistent with the known limits [36]. In fact, as mentioned above, they are already settled to the asymptotic values at k 0 /m ∼ 1000 as shown in the inset of the figure. The imaginary parts Im[N ] for different modes follow the general trend mentioned earlier for |Re[N − 1]| with the x3 mode being the largest and the y1/z2 being the smallest except around k 0 /m ∼ 1, where some crossings occur.
The imaginary part of the refractive index in the weak-field or low-energy (region (2) in Fig. 2) was considered in [37,38]. Although we cannot obtain the analytic expression, we try to compute the imaginary part numerically in this regime. The results are displayed in Fig. 7 for the x2 and x3 modes in the range of 0.03 k 0 /m ≤ 1. The lines labeled as ax2 and ax3 are the results obtained in [37], which are expressed as where = f /f c and ν = k 0 /m. It is found that the imaginary parts Im[N ] are better approximated in this regime by Eqs. (62) and (63) rather than by obtained in [38]. Next we show the dependence of the refractive index on the external-field strength, setting k 0 /m = 1. This has never been published in the literature before. In Fig. 8 56) and (57), labeled as ax2 and ax3 for the x2 and x3 modes, respectively, in the weak-field or low-energy limits.
observed for 0.01 ≤ f /f c 0.5 is in accord with the weak-field or low-energy limits [36], which are given as ax2 and ax3 for the x2 and x3 modes in the inset of this figure, respectively. Re[N − 1] is negative at f /f c 10, which is consistent with the earlier findings. The modulus |Re[N − 1]| is an increasing function of f at f /f c 10.
The imaginary part Im[N ] is shown in Fig. 10. It increases monotonically with the externalfield strength. The slopes are steeper at f /f c 0.5, which is consistent with the analytic expression in the weak-field or low-energy limit of Im[N ] [37,38]. The inset of this figure shows the comparison of our numerical results with the asymptotic limits, Eqs. (62) and (63), labeled as ax2 and ax3 for the x2 and x3 modes, respectively. They almost coincide with each other at f /f c 0.5. Note, on the other hand, that the behavior of the imaginary part at high field-strengths has not been reported in the literature.

Plane-Wave
We next consider the "local" refractive index for the plane wave field, which is also original in this paper. We evaluate numerically the polarization tensor is given in Eqs. (50), (52)-(54) and solve the Maxwell equation, Eq. (55), obtained in the gradient expansion. Since our formulation is based on the perturbation theory, it is natural to express the refractive index in the plane wave as N + δN , where N is the refractive index for the crossed field and δN is the correction from the temporal and spatial non-uniformities. As mentioned for the crossed field, the refractive indices for the y1 and z2 modes are identical to each other. In fact, the relevant components of the Maxwell equations, Eq. (55), are the same for these modes. This is also true for the y3 and z1 modes.
It is found that the correction δN starts indeed with the linear order of Ω/m for both the real and imaginary part. It is then written as and the numerical values of the coefficients C Re and C Im are given for k 0 /m = 1 and f /f c = 1 in Table 2. The temporal and spatial variations are found to mainly affect the imaginary 23/50   Fig. 11. It is seen that the real part can be both positive and negative: it tends to be negative at higher values of k 0 /m although the range depends on the mode; in fact, the values of the photon energy, above which δN The imaginary part Im[δN ] is shown in Fig. 12 for the probe-photon energies of 0.03 ≤ k 0 /m ≤ 1. The inset indicates the comparison for the x2 and x3 modes between the numerically computed results of Im[δN ] and the asymptotic values in the weak-field or low-energy limits. The expressions of Im [δN ] in this regime are given from Eqs. (52) and (53) as Ωk µ n µ m 2 Ωk µ n µ m 2 where Ωk µ n µ /m 2 = Ω(k 0 + k 1 )/m 2 is the product of the momentum of the external planewave and that of the probe photon normalized by the electron mass and is a representative term in the gradient expansion F µν ∼ f µν (1 + Ωξ), being proportional to Ω with the proportional factor k µ n µ originating from the commutation relation of ξ that accompanies Ω; κ 2 = e 2 k µ f µ ν f νλ k λ /m 6 = e 2 f 2 (k 0 + k 1 ) 2 /m 6 as previously defined in Eq. (56). Equations (67) and (68) are convenient for the evaluation of the typical value of Im[δN ]: where I is the intensity of the external electromagnetic wave. It is found from the inset that Im[δN ] is well approximated for Ω/m = 10 −3 by the asymptotic expressions at k 0 /m ≤ 0.03 for f /f c = 1. There occurs a dent at k 0 /m 0.2 and Im[δN ] rises more rapidly with k 0 /m at larger energies, where Im[N ] of the crossed field also becomes substantial. The location of the dent depends on the propagation direction of the probe photon, with the x (y/z) mode having the smallest (largest) value of k 0 /m at the dent, respectively. Since the imaginary part of the refractive index declines rapidly below these energies for the crossed field, it is dominated by the first-order correction Im[δN ] from the temporal and spatial variations in the plane-wave at these low energies. In fact, the latter is commonly more than 10 times larger than the former Im[δN ] 10 × Im[N ] at k 0 /m 0.1. See also Fig. 13, where we plot Im[N + δN ] and Im[N ] as a function of k 0 /m. This is especially the case of the probe photons propagating transversally to the background plane-wave. In accordance with the trend for the crossed field, the x (y/z) modes have largest (smallest) moduli |Im[δN ]| and s modes come in between in general.
Finally, we look into the dependence of δN on the field strength in the range of f /f c ≤ 1. The real and imaginary parts of δN are shown in Figs. 14 and 15, respectively. The probephoton energy is set to k 0 /m = 1 and the frequency of the external field assumed to be 26/50 for different modes is the same as that in Fig. 11; Re[δN ] is negative at a certain range of f /f c , which depends on the mode, occurring for stronger fields for the mode propagating transversally to the background plane-field. The reason for these behaviors is the following: although δN depends not only on the product of k 0 /m and (f /f c ) 2 but also on k µ k µ /m 2 , the latter dependence is minuscule in the regime we consider here. As a result, the dependence of the refractive index on k 0 /m can be translated into that of f /f c . In fact, the numerical results for Im[δN ] are well-approximated by the same asymptotic formulae, Eqs. (67) and (68), in the weak-field regime f /f c 0.1, which can be seen in the inset of Fig. 15

Summary and Discussion
In this paper we have developed a perturbation theory adapted to Schwinger's proper-time method to calculate the induced electromagnetic current, which should be plugged into the Maxwell equations to obtain the refractive indices, for the external, linearly polarized plane-waves, considering them as the unperturbed states and regarding a probe photon as the perturbation to them. Although this is nothing new and indeed was already employed previously [20], our formulation is based on the interaction picture, a familiar tool in quantum mechanics and referred to also as the Furry picture in strong-field QED, rather than utilizing the properties of particular electromagnetic fields from the beginning. Moreover, assuming that the wavelength of the external plane-wave is much longer than the Compton wavelength of electron and employing the gradient expansion, we have evaluated locally the polarization tensor via the induced electromagnetic current to the lowest order of the spatial and temporal 27/50  67) and (68)) labeled as ax2 and ax3, respectively, and the numerically computed results for the x2 and x3 modes labeled as x2 and x3, respectively. variations of the external fields, which is the main achievement in this paper. It has been shown that the vacuum polarization is given locally by the field strength and its gradient of the external plane-waves at each point. We have then considered the dispersion relations for the probe photons propagating in various directions and derived the local refractive indices. We have first evaluated them for the crossed fields, which are the long-wavelength limit of the plane-waves. In so doing, the field strength and the energy of the probe photon are not limited but are allowed to take any values. Note that even for the crossed field not all the parameter regime has been investigated and we have explored those portions unconsidered so far. We have shown that the refractive index is larger for the photons propagating oppositely to the external field than for those propagating perpendicularly. We have also confirmed some 28/50 limiting cases that were already known in the literature analytically or numerically [35][36][37][38], particularly the behavior in the weak external fields demonstrated in [17]. Note, however, that the assumption of a fixed classical background field becomes rather questionable at field strengths near or, in particular, above the critical field strength, since the back reactions to the background field from pair creations should be then taken into account. This issue is certainly much beyond the scope of this paper and in spite of this conceptual problem we think that the results in such very strong fields are still useful to understand the scale and qualitative behavior of the corrections from the field gradient.
We have then proceeded to the evaluation of the refractive index for the plane-wave to the lowest order of the temporal and spatial variations of the background field. The local correction δN to the refractive index for the crossed field N has been numerically evaluated for the first time. We have demonstrated that the modulus of its imaginary part is larger than that of the real part, i.e., the field variations mainly affect the imaginary part of the refractive index. Note that the refractive index we have obtained in this study is local, depending on the local field-strength and its gradient, and is meaningful in the sense of the WKB approximation. This is in contrast to the refractive index averaged over the photon path in [13].
In the optical laser experiments (Ω/m ∼ 10 −6 ), the refractive index may be approximated very well by that for the crossed field. The correction from the field variations is typically |δN | ∼ 10 −5 × Re[N − 1]. The weak-field limit may be also justified, since the current maximum laser-intensity ∼ 2.0 × 10 22 W/cm 2 is still much lower than the critical value, 4.6 × 10 29 W/cm 2 . Then the numerical results given in Fig. 9 are applicable: Re[N − 1] ∼ 10 −4 × (f /f c ) 2 for the probe photon with k 0 /m = 1, which corresponds to ∼ 10 −8 at ∼ 10 25 W/cm 2 , the power expected for future laser facilities such as ELI. Note that how to observe the local refractive index in the electromagnetic wave is a different issue and the averaged one will be better suited for experiments [13].
Unlike for the optical laser, the field variations may not be ignored for x-ray lasers with Ω/m 10 −2 . We find from Fig. 9 and Table 2 that the refractive index for the crossed field and the first-order correction to it are |N − 1| ∼ 10 −4 and |δN | ∼ 10 −5 , respectively, for the probe photon with k 0 /m = 1 propagating oppositely to the external fields with the critical field strength. It may be more interesting that the imaginary part of the first-order correction, Im[δN ], becomes larger than that for the crossed field Im[N ] at f /f c 0.1 for k 0 /m = 1 or at k 0 /m 0.1 for f /f c = 1. It should be noted, however, that the suppression is much relaxed by the presence of the temporal and spatial variations in the background planefield. This is because the imaginary part of the refractive index is exponentially suppressed for the crossed-field while it is suppressed only by powers for the plane-wave.
Very strong electromagnetic fields and their temporal and/or spatial variations may be also important for some astronomical phenomena. For example, burst activities called giant flares and short bursts have been observed in magnetars, i.e., strongly magnetized neutron stars [3]. Although the energy source of these activities is thought to be the magnetic fields of magnetars, the mechanism of bursts is not understood yet. In the analysis of the properties of the emissions from these bursts, the results obtained in this paper may be useful. 29/50 As for the burst mechanism, one interesting model related with the strong field variation was proposed by some authors [50][51][52], in which they considered shock formations in electromagnetic waves propagating in strong magnetic fields around the magnetar. The shock dissipation may produce a fireball of electrons and positrons via pair creations. Their discussion is based on the Rankine-Hugoniot-type jump condition and the Euler-Heisenberg Lagrangian, which is certainly not able to treat the close vicinity of the shock wave, since the shock is essentially a discontinuity. Note, however, that our result in this paper is not very helpful for this problem, either, since the field variation is very rapid and has quite short wavelengths and, moreover, finite amplitudes of waves are essential for shock formation while our method is limited to the linear level. It is hence needed to extend the formulation to accommodate these nonlinear effects somehow, which will be a future task.